Robust predictive deconvolution system and method

ABSTRACT

A method for processing a received, modulated pulse (i.e. waveform) that requires predictive deconvolution to resolve a scatterer from noise and other scatterers includes receiving a return signal; obtaining L+(2M−1)(N−1) samples y of the return signal, where y(l)={tilde over (x)} T (l)s+v(l); applying RMMSE estimation to each successive N samples to obtain initial impulse response estimates [{circumflex over (x)} 1 {−(M−1)(N−1)}, . . . , {circumflex over (x)} 1 {−1}, {circumflex over (x)} 1 {0}, . . . , {circumflex over (x)} 1 {L−1}, {circumflex over (x)} 1 {L}, . . . , {circumflex over (x)} 1 {L−1+(M−1)(N−1)}]; computing power estimates {circumflex over (ρ)} 1 (l)=|{circumflex over (x)} 1 (l)| 2  for l=−(M−1)(N−1), . . . , L−1+(M−1)(N−1); computing MMSE filters according to w(l)=ρ(l)(C(l)+R) −1 s, where ρ(l)=|x(l)| 2  is the power of x(l), and R=E[v(l)v H (l)] is the noise covariance matrix; applying the MMSE filters to y to obtain [{circumflex over (x)} 2 {−(M−2)(N−1)}, . . . , {circumflex over (x)} 2 {−1}, {circumflex over (x)} 2 {0}, . . . , {circumflex over (x)} 2 {L−1}, {circumflex over (x)} 2 {L}, . . . , {circumflex over (x)} 2 {L−1+(M−2)(N−1)}]; and repeating (d)-(f) for subsequent reiterative stages until a desired length-L range window is reached, thereby resolving the scatterer from noise and other scatterers. The RMMSE predictive deconvolution approach provides high-fidelity impulse response estimation. The RMMSE estimator can reiteratively estimate the MMSE filter for each specific impulse response coefficient by mitigating the interference from neighboring coefficients that is a result of the temporal (i.e. spatial) extent of the transmitted waveform. The result is a robust estimator that adaptively eliminates the spatial ambiguities that occur when a fixed receiver filter is used.

The present application claims the benefit of the priority filing dateof provisional patent application No. 60/499,372, filed Sep. 3, 2003,incorporated herein by reference.

TECHNICAL FIELD

This invention relates to a method and system for predictivedeconvolution, which is otherwise known as pulse compression in radarapplications. More particularly, the invention relates to robustpredictive deconvolution using minimum mean-square error reiteration.

BACKGROUND ART

In many sensing applications, such as radar pulse compression, sonar,ultrasonic non-destructive evaluation for structural integrity,biomedical imaging, and seismic estimation, it is desirable to estimatethe impulse response of an unknown system by driving the system with aknown signal having a finite temporal extent. The process of separatingthe known signal from the received output of the system in order toestimate the unknown impulse response is known as predictivedeconvolution.

Predictive deconvolution provides a means to obtain the high spatialresolution of a short, high bandwidth pulse without the need for veryhigh peak transmit power, which may not be feasible. This isaccomplished by transmitting a longer pulse that is phase or frequencymodulated to generate a wideband waveform. The transmission of thewideband waveform into the unknown system results in a received returnsignal at the sensor that is the convolution of the waveform and thesystem impulse response, which possesses large coefficient values atsample delays corresponding to the round-trip travel time of thetransmitted waveform from the sensor to a significant reflecting object(also called a scatterer) and back to the sensor. The purpose ofpredictive deconvolution is to accurately estimate the unknown systemimpulse response from the received return signal based upon the knowntransmitted waveform.

A well-known approach to predictive deconvolution, used extensively inradar and biomedical imaging applications, is known as matchedfiltering, e.g. as described in M. I. Skolnik, Introduction to RadarSystems, McGraw-Hill, New York, 1980, pp. 420-434; T. X. Misaridis, K.Gammelmark, C. H. Jorgensen, N. Lindberg, A. H. Thomsen, M. H. Pedersen,and J. A. Jensen, “Potential of coded excitation in medical ultrasoundimaging,” Ultrasonics, Vol. 38, pp. 183-189, 2000. Matched filtering hasbeen shown to maximize the received signal-to-noise ratio (SNR) in thepresence of white Gaussian noise by convolving the transmitted signalwith the received radar return signal. One can represent matchedfiltering in the digital domain as the filtering operation{circumflex over (x)} _(MF)(l)=s ^(H) {tilde over (y)}(l),  (1)where {circumflex over (x)}_(MF)(l), for l=0, . . . , L−1, is theestimate of the l^(th) delayed sample of the system impulse response,s=[s₁ s₂ . . . s_(N)]^(T) is the length-N sampled version of thetransmitted waveform, {tilde over (y)}(l)=[y(l) y(l+1) . . .y(l+N−1)]^(T) is a vector of N contiguous samples of the received returnsignal, and (•)^(H) and (•)^(T) are the conjugate transpose (orHermitian) and transpose operations, respectively. Each individualsample of the return signal can be expressed asy(l)={tilde over (x)} ^(T)(l)s+v(l),  (2)where {tilde over (x)}(l)=[x(l) x(l−1) . . . x(l−N+1)]^(T) consists ofsamples of the true system impulse response and v(l) is additive noise.The matched filter output can therefore be written as{circumflex over (x)} _(MF)(l)=s ^(H) A ^(T)(l)s+s ^(H) v(l),  (3)where v(l)=[v(l) v(l+1) . . . v(l+N−1)]^(T) and $\begin{matrix}{{A(l)} = \begin{bmatrix}{x(l)} & {x\left( {l + 1} \right)} & \ldots & {x\left( {l + N - 1} \right)} \\{x\left( {l - 1} \right)} & {x(l)} & ⋰ & \vdots \\\vdots & ⋰ & ⋰ & {x\left( {l + 1} \right)} \\{x\left( {l - N + 1} \right)} & \ldots & {x\left( {l - 1} \right)} & {x(l)}\end{bmatrix}} & (4)\end{matrix}$is a collection of sample-shifted snapshots (in the columns) of theimpulse response.

From (4), it is obvious that estimation via matched filtering willsuffer from spatial ambiguities (also known as range sidelobes in theradar vernacular) due to the influence from neighboring impulse responsecoefficients (i.e. the off-diagonal elements of A(l)). To alleviate thiseffect, Least-Squares (LS) solutions have been proposed, e.g. in M. H.Ackroyd and F. Ghani, “Optimum mismatched filters for sidelobesuppression,” IEEE Trans. Aerospace and Electronic Systems, Vol. AES-9,pp. 214-218, March 1973; T. Felhauer, “Digital Signal Processing forOptimum Wideband Channel Estimation in the Presence of Noise,” IEEProceedings-F, Vol. 140, No. 3, pp. 179-186, June 1993; S. M. Song, W.M. Kim, D. Park, and Y. Kim, “Estimation theoretic approach for radarpulse compression processing and its optimal codes,” Electronic Letters,Vol. 36, No. 3, pp. 250-252, February 2000; B. Zrnic, A. Zejak, A.Petrovic, and I. Simic, “Range sidelobe suppression for pulsecompression radars utilizing modified RLS algorithm,” Proc. IEEE Int.Symp. Spread Spectrum Techniques and Applications, Vol. 3, pp.1008-1011, September 1998; and T. K. Sarkar and R. D. Brown, “Anultra-low sidelobe pulse compression technique for high performanceradar systems,” in Proc. IEEE National Radar Conf., pp. 111-114, May1997. LS solutions decouple neighboring impulse response coefficientswhich have been smeared together due to the temporal (and hence spatial)extent of the transmitted waveform. The LS solution models thelength-(L+N−1) received return signal asy=Sx+v,  (5)where x=[x(0) x(1) . . . x(L−1)]^(T) are the L true impulse responsecoefficients that fall within the data window, v=[v(0) v(1) . . .v(L+N−2)]^(T) are additive noise samples, and the convolution of thetransmitted waveform with the system impulse response is approximated asthe matrix multiplication $\begin{matrix}{{S\quad x} = {\begin{bmatrix}s_{1} & 0 & \ldots & 0 \\\vdots & s_{1} & \quad & \vdots \\s_{N} & \vdots & ⋰ & 0 \\0 & s_{N} & \quad & s_{1} \\\vdots & \quad & ⋰ & \vdots \\0 & \ldots & 0 & s_{N}\end{bmatrix}{x.}}} & (6)\end{matrix}$The LS model of (6) is employed extensively in radar pulse compression,seismic estimation, e.g. as described in R. Yarlaggadda, J. B. Bednar,and T. L. Watt, “Fast algorithm for l_(p) deconvolution,” IEEE Trans.Acoustics, Speech, and Signal Processing, Vol. ASSP-33, No. 1, pp.174-182, February 1985; and ultrasonic non-destructive evaluation, e.g.as in M. S. O'Brien, A. N. Sinclair, and S. M. Kramer, “High resolutiondeconvolution using least-absolute-values minimization,” Proc.Ultrasonics Symposium, pp. 1151-1156, December 1990; and D.-M. Suh,W.-W. Kim, and J.-G. Chung, “Ultrasonic inspection of studs (bolts)using dynamic predictive deconvolution and wave shaping,” IEEE Trans.Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 46, No. 2, pp.457-463, March 1999. The general form of the LS solution is{circumflex over (x)} _(LS)=(S ^(H) S)⁻¹ S ^(H) y.  (7)

For the received signal model of (5), it can be shown that the LSsolution of (7) is optimal in the mean-square error (MSE) sense when theadditive noise is white. However, upon further inspection one finds thatthe LS received signal model does not completely characterize thereceived return signal because it does not account for the convolutionof the transmitted waveform with impulse response coefficients x(l)prior to l=0. The result is that the presence of a significant impulseresponse coefficient within N−1 samples prior to x(0) can cause severemis-estimation of the desired coefficients within the data window.

There is, therefore, a need for a predictive deconvolution system withimproved robustness, accuracy, and resolution.

DISCLOSURE OF THE INVENTION

According to the invention, a method for processing a received,modulated pulse that requires predictive deconvolution to resolve ascatterer from noise and other scatterers includes (a) receiving areturn signal; (b) obtaining L+(2M−1)(N−1) samples y of the returnsignal, where y(l)={tilde over (x)}^(T)(l)s+v(l); (c) applying RMMSEestimation to each set of N contiguous samples to obtain initial impulseresponse estimates [{circumflex over (x)}₁{−(M−1)(N−1)}, . . . ,{circumflex over (x)}₁{−1}, {circumflex over (x)}₁{0}, . . . ,{circumflex over (x)}₁{L−1}, {circumflex over (x)}₁{L}, . . . ,{circumflex over (x)}₁{L−1+(M−1)(N−1)}]; (d) computing power estimates{circumflex over (ρ)}₁(l)=|{circumflex over (x)}₁(l)|² forl=−(M−1)(N−1), . . . , L−1+(M−1)(N−1); (e) computing MMSE filtersaccording to w(l)=ρ(l)(C(l)+R)⁻¹s, where ρ(l)=|x(l)|² is the power ofx(l), and R=E[v(l) v^(H)(l)] is the noise covariance matrix; (f)applying the MMSE filters to y to obtain [{circumflex over(x)}₂{−(M−2)(N−1)}, . . . , {circumflex over (x)}₂{−1}, {circumflex over(x)}₂{0}, . . . , {circumflex over (x)}₂{L−1}, {circumflex over(x)}₂{L}, . . . , {circumflex over (x)}₂{L−1+(M−2)(N−1)}]; and repeating(d)-(f) for subsequent reiterative stages until a desired length-L rangewindow is reached, thereby resolving the scatterer from noise and otherscatterers.

Also according to the invention, a radar receiver system includes areceiver, a processor including a RMMSE radar pulse compressionalgorithm, and a target detector.

Also according to the invention, a method for processing a received,modulated radar pulse to resolve a radar target from noise or othertargets includes (a) receiving a radar return signal; (b) obtainingL+(2M−1)(N−1) samples y of the radar return signal, where y(l)={tildeover (x)}^(T)(l)s+v(l); (c) applying RMMSE pulse compression to each setof N contiguous samples to obtain initial radar impulse responseestimates [{circumflex over (x)}₁{−(M−1)(N−1)}, . . . , {circumflex over(x)}₁{−1}, {circumflex over (x)}₁{0}, . . . , {circumflex over(x)}₁{L−1}, {circumflex over (x)}₁{L}, . . . , {circumflex over(x)}₁{L−1+(M−1)(N−1)}]; (d) computing power estimates {circumflex over(ρ)}₁(l)=|{circumflex over (x)}₁(l)|² for l=−(M−1)(N−1), . . . ,L−1+(M−1)(N−1); (e) computing range-dependent filters according tow(l)=ρ(l)(C(l)+R)⁻¹s, where ρ(l)=|x(l)|² is the power of x(l), andR=E[v(l) v^(H)(l)] is the noise covariance matrix; (f) applying therange-dependent filters to y to obtain [{circumflex over(x)}₂{−(M−2)(N−1)}, . . . , {circumflex over (x)}₂{−1}, {circumflex over(x)}₂{0}, . . . , {circumflex over (x)}₂{L−1}, {circumflex over(x)}₂{L}, . . . , {circumflex over (x)}₂{L−1+(M−2)(N−1)}]; and repeating(d)-(f) for subsequent reiterative stages until a desired length-L rangewindow is reached, thereby resolving the radar target from noise orother targets.

The Reiterative Minimum Mean-Square Error (RMMSE) predictivedeconvolution system and method of the invention provide high-fidelityimpulse response estimation compared to other systems such as matchedfiltering and LS. In one embodiment, the RMMSE estimator reiterativelyestimates the MMSE filter for each specific impulse response coefficientby mitigating the interference from neighboring coefficients that is aresult of the temporal (i.e. spatial) extent of the transmittedwaveform. The result is a robust estimator that adaptively eliminatesthe spatial ambiguities that occur when a fixed receiver filter is used.

The invention has obvious applications in military radar and sonar, andit is also useful for civilian radar, e.g. in airport radar systems andin weather radar systems, e.g. Doppler radar, and other environmentalradar applications. It may also find use in range profiling, imagerecognition for Synthetic Aperture Radar (SAR) and Inverse SAR (ISAR),remote sensing, ultrasonic non-destructive evaluation for structuralintegrity, seismic estimation, biomedical imaging, inverse filtering ofoptical images, or any other application requiring robust deconvolutionof a known waveform (or filter) from a desired unknown impulse response.

Additional features and advantages of the present invention will be setforth in, or be apparent from, the detailed description of preferredembodiments which follows.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a predictive deconvolution systemaccording to the invention.

FIG. 2 is a block diagram of a 3-stage RMMSE predictive deconvolutionalgorithm according to the invention.

FIG. 3 is a graph showing an impulse response with a single largecoefficient (scatterer) in the data window according to the invention.

FIG. 4 is a graph comparing the performance of the RMMSE algorithmaccording to the invention with LS and matched filter systems for asingle-scatterer impulse response.

FIG. 5 is a graph as in FIG. 3 but showing an impulse response with asecond large coefficient just prior to the data window according to theinvention.

FIG. 6 is a graph comparing the RMMSE, LS, and matched filter systemsfor a scatterer just prior to the data window.

FIG. 7 is a graph comparing the RMMSE, LS, and the matched filtersystems for a realistic scenario in which there are several scattererswith substantially varying magnitudes.

BEST MODE FOR CARRYING OUT THE INVENTION

Definitions: The term “convolution” means the process that yields theoutput response of an input to a linear time-invariant system, and inthe general discrete sense, an input x(n) is convolved with a LinearTime Invariant (LTI) system h(n) to yield an output y(n) as${y(n)} = {\sum\limits_{k = {- \infty}}^{k = \infty}{{x(k)}{h\left( {n - k} \right)}}}$such as is described and defined in J. G. Proakis and D. G. Manolakis,Digital Signal Processing: Principles, Algorithms, and Applications, 3rdEd., pp. 75-82, Prentice Hall: Upper Saddle River, N.J. (1996). The term“deconvolution” as used herein means the process that given the outputof a system determines an unknown input signal to the system. Given anoutput y(n) that is the convolution of some input x(n) to some LTIsystem h(n), deconvolution is the inverse operation that takes y(n) andproduces x(n). Deconvolution is a general term meaning that x(n) is tobe estimated from the observed y(n). See Id. at p. 355. The term“scatterer” means something in the path of a transmitted waveform thatcauses a significant reflection (relative to the noise) back to thereceiver of the sensor.

Referring now to FIG. 1, a predictive deconvolution system 10 includes atransmitter 12 for transmitting a phase or frequency modulated pulse (orwaveform) 14 that, upon interacting with its transmission environment'sunknown impulse response 16 (to be estimated, as is described furtherbelow) and forms a signal 18 that is a convolution of waveform 14 andimpulse response 16, an analog front-end receiver 18 for receivingsignal 16, an optional analog-to-digital (A/D) converter 20, a processor22 for processing analog or digital signal 16, and a detector 24 Theprocessor 22 includes a Reiterative Minimum Mean-Square Error (RMMSE)estimation algorithm, as is described below.

Minimum Mean-Square Error (MMSE) estimation is a Bayesian estimationapproach that employs prior information in order to improve estimationaccuracy. The exact form that the prior information will take will beaddressed shortly. First, however, the signal model must be constructed.From (3), we see that the collection of N samples of the received returnsignal can be expressed as{tilde over (y)}(l)=A ^(T)(l)s+v(l).  (8)

This is the received signal model used by the matched filter formulationand takes into account all the necessary impulse response coefficientsfor a given received return sample. To develop the MMSE filter, thematched filter s^(H) in (3) is replaced with the MMSE filter, denotedw^(H)(l), in which the form of the MMSE filter is dependent upon thelength-N sample swath of the impulse response under consideration.Thereafter, the MMSE cost functionJ(l)=E[|x(l)−w ^(H)(l){tilde over (y)}(l)|²]  (9)is solved for each impulse response coefficient l=0, . . . , L−1, whereE[•] denotes expectation. This is done by differentiating with respectto w^(H)(l) and then setting the result to zero. The MMSE filter isfound to take the formw(l)=(E[{tilde over (y)}(l){tilde over (y)} ^(H)(l)])⁻¹ E[{tilde over(y)}(l)x*(l)],  (10)where (•)* is the complex conjugate operation. After substituting for{tilde over (y)}(l) from (8) and assuming that the impulse responsecoefficients are, in general, uncorrelated with one another and are alsouncorrelated with the noise, one obtainsw(l)=ρ(l)(C(l)+R)⁻¹ s,  (11)where ρ(l)=E[|x(l)|²] is the expected power of x(l), and R=E[v(l)v^(H)(l)] is the noise covariance matrix. On assuming neighboringcoefficients are uncorrelated, the (i,j)^(th) element of the matrix C(l)is $\begin{matrix}{{C(l)}_{i,j} = {E\left\lbrack {\sum\limits_{n = \kappa_{L}}^{\kappa_{U}}{{\rho\left( {l - n + i - 1} \right)}{s(n)}{s^{*}\left( {n - i + j} \right)}}} \right\rbrack}} & (12)\end{matrix}$in which κ_(L)=max{0,i−j} is the summation lower bound andκ_(U)=min{N−1,N−1+i−j} is the upper bound. Also, any prior informationregarding the noise can be employed via the noise covariance matrix R.For instance, for a white noise assumption R is diagonal.

In its current state the MMSE filter is a function of the powers of thesurrounding impulse response coefficients, which in practice areunavailable. This lack of prior knowledge can be taken into account bysetting all the initial coefficient estimates equal. Therefore, theinitial MMSE filter reduces to the form

 {tilde over (w)}≅{tilde over (C)}⁻¹s  (13)

where the noise term is assumed negligible and $\begin{matrix}{{\overset{\sim}{C}}_{i,j} = \left\lbrack {\sum\limits_{n = \kappa_{L}}^{\kappa_{U}}{{s(n)}{s^{*}\left( {n - i + j} \right)}}} \right\rbrack} & (14)\end{matrix}$

is invariant to the sample delay l. The initial MMSE filter cantherefore be pre-computed and then implemented in the same way as thetraditional matched filter. The inclusion of the matrix {tilde over (C)}serves to provide a “local” LS initial estimate.

FIG. 2 illustrates the Reiterative MMSE (RMMSE) predictive deconvolutionalgorithm for three iterations. In general, the RMMSE algorithm operatesas follows:

-   1) Collect the L+(2M−1)(N−1) samples of the received return signal    [y{−(M−1)(N−1)}, . . . , y{−1}, y{0}, . . . , y{L−1}, y{L}, . . . ,    y{L−1+M(N−1)}], which comprise the length-L data window along with    the (M−1)(N−1) samples prior to the data window and the M(N−1)    samples after the data window.-   2) Apply the initial MMSE filter from (13) to obtain the initial    impulse response estimates [{circumflex over (x)}₁{−(M−1)(N−1)}, . .    . , {circumflex over (x)}₁{−1}, {circumflex over (x)}₁{0}, . . . ,    {circumflex over (x)}₁{L−1}, {circumflex over (x)}₁{L}, . . . ,    {circumflex over (x)}₁{L−1+(M−1)(N−1)}].-   3) Compute the initial power estimates {circumflex over    (ρ)}₁(l)=|{circumflex over (x)}₁(l)|² for l=−(M−1)(N−1), . . . ,    L−1+(M−1)(N−1) which are used to compute the filters w₁(l) as in    (11), and then apply to {tilde over (y)}(l) to obtain [{circumflex    over (x)}₂{−(M−2)(N−1)}, . . . , {circumflex over (x)}₂{−1},    {circumflex over (x)}₂{0}, . . . , {circumflex over (x)}₂{L−1},    {circumflex over (x)}₂{L}, . . . , {circumflex over    (x)}₂{L−1+(M−2)(N−1)}].-   4) Repeat 2) and 3), changing the indices where appropriate, until    the desired length-L data window is reached.

The initial estimate of the impulse response found by applying the MMSEfilter is used as a priori information to reiterate the MMSE filter andimprove performance. This is done by employing the MMSE filterformulation from (11) in which the respective powers of the impulseresponse coefficients are taken from the current estimate. It has beenfound that two or three reiteration steps allow the RMMSE filter toexclude the effects of scatterers prior to the data window, as well asto suppress the spatial ambiguities very close to the level of the noisefloor. The RMMSE filter does especially well when the impulse responseis somewhat sparsely parameterized (i.e. highly spiky), as is the casefor system designed to have high resolution. Note that each reiterationstep will reduce the number of coefficient estimates by 2(N−1). Tocounteract this, it is necessary to increase the size of the data windowby 2M (N−1) samples, where M is the number of reiteration steps.Typically, however, L>>N so that this reduction in data window size isnegligible.

An important factor regarding practical implementation is thenon-singularity of the N×N matrix ([C(l)]+[R]). This can be addressed byinstituting a nominal level for which the estimated coefficients are notallowed to go below. An alternative to this would be to re-estimate onlythose coefficients that are above some threshold since the small-valuedcoefficients do not contain a detectable scatterer.

To demonstrate the performance of the RMMSE algorithm we examine threecases. The first case is typical of the scenario often addressed for LSestimation techniques and consists of an impulse response with a singlelarge scatterer in noise and low-level clutter. The second case weaddress is when there is a second large scatterer that resides justprior to the data window. In this case it is expected that the LSestimator will substantially degrade since this important region is notaccounted for in the received signal model. The final case is a morerealistic scenario in which there are several spatially dispersedscatterers of varying magnitude.

For the first two cases, the waveform selected is the length N=30polyphase modulated Lewis-Kretschmer P3 code, which on receive (afterdown-conversion to baseband in the receiver front-end) is defined as$\begin{matrix}{{{s(n)} = {\exp\left( \frac{j\quad\pi\quad n^{2}}{N} \right)}},\quad{n = 0},1,\ldots\quad,{N - 1.}} & (15)\end{matrix}$The noise and clutter are modeled as zero-mean Gaussian with powers setto 70 dB and 50 dB below the signal power, respectively. For both caseswe perform two iterations of the RMMSE algorithm and then compare theRMMSE-estimated impulse response with the true impulse response, as wellas with the results obtained from using LS and the matched filter. Theimpulse response for the first case is depicted in FIG. 3 for a singlescatterer present in the data window which consists of 200 range gates.FIG. 4 illustrates the results from the different estimation techniquesin which the matched filter experiences significant spatial ambiguitieswhile both LS and RMMSE have suppressed the spatial ambiguities so thatthe true scatterer location is evident and smaller nearby scattererscould be detectable.

For the scenario just described, the LS and RMMSE estimators performalmost identically. However, when there is a significant scattererpresent just prior to the range window, as depicted in FIG. 5, the LSestimator is expected to degrade substantially. From FIG. 6, the LSestimator truly does suffer from severe mis-estimation when asignificant scatterer cannot be expressed in the model. However, theperformance of the RMMSE estimator is indistinguishable from theprevious case where only a single scatterer was present. This isobviously due to the fact that the RMMSE estimator takes the previousestimates of the surrounding impulse response coefficients into accountwhen estimating the impulse response.

Table 1 presents the Mean Squared-Error (MSE) for both cases discussedusing LS, RMMSE, and normalized matched filtering. The MSE is averagedfor all 200 coefficients in the data window and over 100 runs using thesame scatterer(s), with the clutter and noise distributed according to azero-mean Gaussian distribution for each run independently.

TABLE 1 MSE performance comparison Norm. MF LS RMMSE Case 1 −12.3 dB−40.9 dB −35.8 dB Case 2 −11.2 dB −11.3 dB −34.7 dB

For case 1 (single scatterer), the normalized matched filter has a largeMSE due to spatial ambiguities, while LS and RMMSE perform nearly thesame with LS just marginally better. However, for case 2 (scatterer justprior to the data window), the MSE attained by LS degrades to nearlythat of the normalized matched filter, while the RMMSE maintains roughlythe same MSE as in the previous case.

For the third case, we examine the performance for the three estimationtechniques when the impulse response contains several dispersedscatterers of varying magnitude (both within and outside of the datawindow). FIG. 7 illustrates the significant improvement in estimationperformance of the RMMSE estimator as compared to matched filtering andLS estimation. Most notable is the resolvability of two very closelyspaced scatterers. RMMSE completely resolves the two scatterers whereasfor LS and matched filtering it is not completely clear whether it isone or two scatterers.

Obviously many modifications and variations of the present invention arepossible in the light of the above teachings. It is therefore to beunderstood that the scope of the invention should be determined byreferring to the following appended claims.

1. A method for processing a received, modulated pulse that requirespredictive deconvolution to resolve a scatterer from noise and otherscatterers, comprising: a) receiving a return signal; b) obtainingL+(2M−1)(N−1) samples y of the return signal, where y(l)={tilde over(x)}^(T)(l)s+v(l); c) applying RMMSE estimation to each successive Nsamples to obtain initial impulse response estimates [{circumflex over(x)}₁{−(M−1)(N−1)}, . . . , {circumflex over (x)}₁{−1}, {circumflex over(x)}₁{0}, . . . , {circumflex over (x)}₁{L−1}, {circumflex over(x)}₁{L}, . . . , {circumflex over (x)}₁{L−1+(M−1)(N−1)}]; d) computingpower estimates {circumflex over (ρ)}₁(l)=|{circumflex over (x)}₁(l)|²for l=−(M−1)(N−1), . . . , L−1+(M−1)(N−1); (e) computing MMSE filtersaccording to w(l)=σ(l)(C(l)+R)⁻¹s, where ρ(l)=|x(l)|² is the power ofx(l), and R=E[v(l)v^(H)(l)] is the noise covariance matrix; (f) applyingthe MMSE filters to y to obtain [{circumflex over (x)}₂{−(M−2)(N−1)}, .. . , {circumflex over (x)}₂{−1}, {circumflex over (x)}₂{0}, . . . ,{circumflex over (x)}₂{L−1}, {circumflex over (x)}₂{L}, . . . ,{circumflex over (x)}₂{L−1+(M−2)(N−1)}]; and (g) repeating (d)-(f) forsubsequent reiterative stages until a desired length-L range window isreached, thereby resolving the scatterer from noise and otherscatterers.
 2. A method as in claim 1, wherein the RMMSE estimation isperformed with a plurality of parallel processors.
 3. A method as inclaim 1, further comprising setting a nominal level for which the powerestimates are not allowed to fall below.
 4. A method as in claim 1,wherein the y samples are obtained via A/D conversion.
 5. A method as inclaim 1, wherein the method is applied in range profiling.
 6. A methodas in claim 1, wherein the method is applied in a weather radar system.7. A method as in claim 1, wherein the method is applied in imagerecognition for Synthetic Aperture Radar (SAR).
 8. A method as in claim1, wherein the method is applied in image recognition for Inverse SAR(ISAR).
 9. A method as in claim 1, wherein the method is applied inremote sensing.
 10. A method as in claim 1, wherein the method isapplied in ultrasonic non-destructive evaluation for structuralintegrity.
 11. A method as in claim 1, wherein the method is applied inseismic estimation.
 12. A method as in claim 1, wherein the method isapplied in biomedical imaging.
 13. A method as in claim 1, wherein themethod is applied in inverse filtering of optical images.
 14. A radarreceiver system, comprising: a receiver; a processor including aReiterative Minimum Mean-Square Error estimation (RMMSE) radar pulsecompression algorithm; and a target detector.
 15. A radar receiversystem as in claim 14, wherein the RMSSE radar pulse compressionalgorithm comprises: (a) obtaining L+(2M−1)(N−1) samples y of a radarreturn signal, where y(l)={tilde over (x)}^(T)(l)s+v(l); (b) applyingRMMSE pulse compression to each set of N contiguous samples to obtaininitial radar impulse response estimates [{circumflex over(x)}₁{−(M−1)(N−1)}, . . . , {circumflex over (x)}₁{−1}, {circumflex over(x)}₁{0}, . . . , {circumflex over (x)}₁{L−1}, {circumflex over(x)}₁{L}, . . . , {circumflex over (x)}₁{L−1+(M−1)(N−1)}]; (c) computingpower estimates {circumflex over (ρ)}₁(l)=|{circumflex over (x)}₁(l)|²for l=−(M−1)(N−1), . . . , L−1+(M−1)(N−1); (d) computing range-dependentfilters according to w(l)=ρ(l)(C(l)+R)⁻¹s, where ρ(l)=|x(l)|² is thepower of x(l), and R=E[v(l)v^(H)(l)] is the noise covariance matrix; (e)applying the range-dependent filters to y to obtain [{circumflex over(x)}₂{−(M−2)(N−1)}, . . . , {circumflex over (x)}₂{−1}, {circumflex over(x)}₂{0}, . . . , {circumflex over (x)}₂{L−1}, {circumflex over(x)}₂{L}, . . . , {circumflex over (x)}₂{L−1+(M−2)(N−1)}]; and (f)repeating (c)-(e) for subsequent reiterative stages until a desiredlength-L range window is reached.
 16. A radar receiver system as inclaim 14, further comprising a plurality of parallel processors forperforming the RMMSE pulse compression.
 17. A radar receiver system asin claim 14, wherein a nominal level is set for which the powerestimates are not allowed to fall below.
 18. A radar receiver system asin claim 14, further comprising an analog-to-digital (A/D) converter.19. A radar receiver system as in claim 15, further comprising ananalog-to-digital (A/D) converter for obtaining the y samples.
 20. Aradar receiver system as in claim 14, wherein the system is an airportradar system.
 21. A radar receiver system as in claim 14, wherein thesystem is a weather radar system.
 22. A method for processing areceived, modulated radar pulse to resolve a radar target from noise orother targets, comprising: a) receiving a radar return signal; b)obtaining L+(2M−1)(N−1) samples y of the radar return signal, wherey(l)={tilde over (x)}^(T)(l)s+v(l); c) applying RMMSE pulse compressionto each successive N samples to obtain initial radar impulse responseestimates [{circumflex over (x)}₁{−(M−1)(N−1)}, . . . , {circumflex over(x)}₁{−1}, {circumflex over (x)}₁{0}, . . . , {circumflex over(x)}₁{L−1}, {circumflex over (x)}₁{L}, . . . , {circumflex over(x)}₁{L−1+(M−1)(N−1)}]; d) computing power estimates {circumflex over(ρ)}₁(l)=|{circumflex over (x)}₁(l)|² for l=−(M−1)(N−1), . . . ,L−1+(M−1)(N−1); (e) computing range-dependent filters according tow(l)=ρ(l)(C(l)+R)⁻¹s, where ρ(l)=|x(l)|² is the power of x(l), andR=E[v(l)v^(H)(l)] is the noise covariance matrix; (f) applying therange-dependent filters to y to obtain [{circumflex over(x)}₂{−(M−2)(N−1)}, . . . , {circumflex over (x)}₂{−1}, {circumflex over(x)}₂{0}, . . . , {circumflex over (x)}₂{L−1}, {circumflex over(x)}₂{L}, . . . , {circumflex over (x)}₂{L−1+(M−2)(N−1)}]; and (g)repeating (d)-(f) for subsequent reiterative stages until a desiredlength-L range window is reached, thereby resolving the radar targetfrom noise or other targets.
 23. A method as in claim 22, wherein theRMMSE pulse compression is performed with a plurality of parallelprocessors.
 24. A method as in claim 22, wherein the y samples of theradar return signal are obtained via A/D conversion.
 25. A method as inclaim 22, further comprising setting a nominal level for which the powerestimates are not allowed to fall below.
 26. A method as in claim 22,wherein the method is applied in an airport radar system.
 27. A methodas in claim 22, wherein the method is applied in a weather radar system.28. A method as in claim 22, wherein the y samples of the radar returnsignal are obtained via A/D conversion.
 29. A method as in claim 22,wherein a plurality of radar targets are resolved and separatelyidentified.